attractor mechanism as a distillation procedure

Attractor mechanism as a distillation procedure

Peter Levay and Szilard Szalay

Department of Theoretical Physics, Institute of Physics, Budapest University of Technology and Economics,H-1521 Budapest, Hungary

(Received 27 April 2010; published 1 July 2010)

In a recent paper it was shown that for double extremal static spherical symmetric BPS black hole

solutions in the STU model the well-known process of moduli stabilization at the horizon can be recast in

a form of a distillation procedure of a three-qubit entangled state of a Greenberger-Horne-Zeilinger type.

By studying the full flow in moduli space in this paper we investigate this distillation procedure in more

detail. We introduce a three-qubit state with amplitudes depending on the conserved charges, the warp

factor, and the moduli. We show that for the recently discovered non-BPS solutions it is possible to see

how the distillation procedure unfolds itself as we approach the horizon. For the non-BPS seed solutions at

the asymptotically Minkowski region we are starting with a three-qubit state having seven nonequal

nonvanishing amplitudes and finally at the horizon we get a Greenberger-Horne-Zeilinger state with

merely four nonvanishing ones with equal magnitudes. The magnitude of the surviving nonvanishing

amplitudes is proportional to the macroscopic black hole entropy. A systematic study of such attractor

states shows that their properties reflect the structure of the fake superpotential. We also demonstrate that

when starting with the very special values for the moduli corresponding to flat directions the uniform

structure at the horizon deteriorates due to errors generalizing the usual bit flips acting on the qubits of the

attractor states.

DOI: 10.1103/PhysRevD.82.026002 PACS numbers: 11.25.Mj, 03.65.Ud, 03.67.Mn, 04.70.Dy

I. INTRODUCTION

Recently striking multiple relations have been discov-ered between two seemingly unrelated fields: the physicsof black hole solutions in string theory and the theory ofquantum entanglement within quantum information theory[1–3]. Further papers established a complete dictionarybetween a variety of phenomena on one side of the corre-spondence in the language of the other. This black hole-qubit correspondence has repeatedly proved to be usefulfor obtaining additional insight into both of the two fields[4–12]. The main correspondence found [1,2,4–6,13] isbetween the macroscopic entropy formulas obtained forcertain black hole solutions in supergravity theories andmultiqubit and qutrit entanglement measures used in quan-tum information theory (QIT).

These results are intriguing mathematical connectionsarising from the similar symmetry properties of qubitsystems and the web of dualities in string theory. In thecase of the STU model, our main concern here, the sym-metry group in question at the classical level is SLð2;RÞ�3,or taking into account the quantized nature of electric andmagnetic charges it is SLð2;ZÞ�3. In string theory the lattersymmetry group is also dictated by internal consistency. Inqubit systems on the other hand the symmetry group inquestion is the group of stochastic local operations andclassical communication (SLOCC) which is [14]GLð2;CÞ�3, a group which is not changing the entangle-ment type of a three-qubit system. Generally for the casesstudied so far the groups connected to dualities occurringin stringy black holes are related to integers or to the real

numbers, and the ones of QIT are the complex versions ofthe relevant groups.In spite of the fact that most of the results valuable also

on the QIT side were obtained by string theorists there is anencouraging trend to get more theorists involved also fromthe QIT side. The reason for the reluctance from the QITside might be due to the fact that in QIT the symmetrygroups in question are complex (qubits) and on the stringtheory side real (rebits). Bearing in mind this differenceone should notice however, that some of the results ob-tained in string theory still proved to be useful for con-structing nontrivial new entanglement measures in QIT.For example, the quartic invariants of Freudenthal triplesystems over the integers which are well known to stringtheorists when used over the complex numbers gave hintsto construct a genuine measure for solving the classifica-tion problem of three fermions with six single particlestates [15]. These results employing further use ofFreudenthal systems enabled a further classification ofspecial entangled systems containing indistinguishableconstituents of both fermions and bosons [16].The other important impact coming from the string

theory side is associated with the observation that the so-called real Pauli group which is of fundamental importancein the stabilizer formalism of quantum error correctingcodes [17] gives rise to interesting finite geometric struc-tures that can be related to the structure of black holeentropy formulas [13,18]. Such finite geometric structureswere already used in QIT [19,20] for describing the com-mutation algebra of multiqubit systems. In the spirit ofthese papers using the input from string theory it has only

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recently become clear that black hole entropy formulasalso provide a way of obtaining new configurations gen-eralizing the well-known ones of Mermin squares [18],objects known from Bell-Kochen-Specker-like theoremson hidden variables [21]. These studies also emphasizedan important connection between the structure of incidencegeometries and their finite automorphism groups realizedin terms of quantum gates of quantum information theory.In this spirit groups like the Weyl groupsWðE6Þ andWðE7Þand the simple group PSLð2; 7Þ as finite subgroups of theinfinite discrete U-duality group known from string theoryhas been linked to the Clifford group of quantum compu-tation [13,18,22,23].

Apart from understanding black hole entropy in quan-tum information theoretic terms the desire for an entangle-ment based understanding for issues of dynamics ofattractors also arose. In particular, in the special case ofthe STU model [11,24,25] it has been realized [3] that forextremal spherically symmetric BPS black hole solutions itis possible to rephrase the attractor mechanism [26] as adistillation procedure of entangled ‘‘states’’ of a very spe-cial kind on the event horizon. Such states are of a GHZtype [27] or graph states [7,28] well known from quantuminformation theory. The basic tool for establishing thisresult was the introduction of a three-qubit state j�i de-pending on the conserved charges and also on the modulifields. Such states enjoy a number of remarkable properties[7]. The norm of this state having 8 amplitudes is the blackhole potential [11] VBH. The flat covariant derivatives withrespect to the Kahler connection are acting on j�i as bit-flip errors on the qubits. At the horizon bit-flip errors onj�i are suppressed for BPS solutions and for non-BPSones they are not. The non-BPS solutions can be charac-terized by the number and types of bit-flip errors.

However, in these investigations [3,7] establishing theseresults only double extremal [25] solutions have beenconsidered for which the moduli fields are constant evenaway from the horizon. Since for this class of solutions j�iis also constant clearly within the context of such solutionsit is not possible to get any additional insight into theimportant question of how the distillation procedure un-folds itself as we approach the horizon after taking the limitr ! 0 with r being the radial coordinate.

Luckily both in the BPS and non-BPS cases there existmore general static spherically symmetric solutions featur-ing the full radial flow in moduli space. For the BPS casethese are the well-known solutions based on harmonicfunctions [29] and for the non-BPS case similar resultsgeneralizing these ones have recently become available[11,30–32]. In these works after solving the equations ofmotion one obtains the attractor flow zjðrÞ in moduli space.Hence employing the charge and moduli dependent multi-qubit states [3,7] and using these solutions one might hopeto get some additional insight into distillation issues bystudying the corresponding flow j�ðrÞi.

The aim of the present paper is to investigate this dis-tillation procedure in detail for the special case of extremalspherically symmetric black hole solutions in the STUmodel. We will use a special combination of the modulifields, the warp factor, and the conserved charges reminis-cent of a three-qubit state of quantum information theory.We will call this creature a ‘‘three-qubit state’’ furnishing arepresentation space for the action of the duality groupSLð2;RÞ�3 � Spð8;RÞ, though this terminology might bemisleading. It will be obvious that our state has intimateconnections to entities like the ‘‘fake superpotential’’ [33]and even possibly to the phase of the semiclassical wavefunction used in recent studies [34], however in this paperwe will not elaborate on its physical meaning. Results onthe origin of these three-qubit states having some relevanceon such interesting issues will be presented in an accom-panying paper [35].The organization of this paper is as follows. In Sec. II we

summarize the usual formalism of the STU model. InSec. III we introduce our moduli and charge-dependentthree-qubit state, and recall results concerning the blackhole-qubit correspondence that we will need later. InSec. IV we reformulate the well-known findings concern-ing BPS solutions based on harmonic functions. Here weshow that the ‘‘attractor at infinity’’ [11,36] corresponds toa distillation procedure of a normalized GHZ state (dual tothe usual one at the horizon [3]) at the asymptoticallyMinkowski region. In Sec. V we study the flow j�ðrÞifor the non-BPS D0�D4 system answering the seedsolution [31]. Here we generalize further our three-qubitstate by including also the warp factor into its definition.We show that the Fourier amplitudes of this state in thediscrete Fourier (Hadamard) transformed basis satisfy a setof first order differential equations. Using the results of theprevious sections in Sec. VI for the non-BPS seed solutionswe demonstrate how a GHZ state at the horizon emergesfrom a state characterizing the flow at the asymptoticallyMinkowski region. The attractor mechanism in this picturesimply amounts to the fact that three amplitudes out of theseven nonequal nonvanishing ones are dying out as weapproach the horizon. The remaining amplitudes have thesame magnitudes related to the macroscopic black holeentropy. The relative phase factors of these amplitudes aremerely signs reflecting the structure of the fake superpo-tential [11,33]. In Sec. VII using recent general results onthe STU model [11] we calculate the explicit form of thestates at the horizon for both the available BPS and non-BPS solutions. Featuring the parametrization [11,31] re-vealing the flat directions [37] we show that the role of theflat directions in this picture is to deteriorate the uniformGHZ-like structure on the horizon. According to the resultsof our previous paper [7] the differences between differenttypes of solutions (BPS, non-BPS with vanishing [8] andnonvanishing [38] central charge) manifest themselves inapplying bit-flip errors to the relevant GHZ-like state of the

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BPS flow. In view of this result flat directions give a newtwist on this picture, namely, when starting the flow in oneof the flat directions the resulting state on the horizon willexhibit errors of a more general type than the usual bit-flipones. Finally in Sec. VIII we present our conclusions, withsome calculational details left to the Appendix.

II. STU BLACK HOLES

In the following we consider ungauged N ¼ 2 super-gravity in d ¼ 4 coupled to n vector multiplets. The n ¼ 3case corresponds to the STU model. The bosonic part ofthe action (without hypermultiplets) (in units ofGN ¼ 1) is

S ¼ 1

16�

Zd4x

ffiffiffiffiffiffijgj

q ��R

2þGa �b@�z

a@� �z�bg��

þ ðImN IJF I �F J þ ReN IJF I � �F JÞ�: (1)

Here F I, and �F I, I ¼ 0; 1; 2; . . . ; n are two-forms asso-ciated to the field strengthsF I

�� of nþ 1U(1) gauge fields

and their duals. The za a ¼ 1; . . . ; n are complex scalar(moduli) fields that can be regarded as local coordinates ona projective special Kahler manifoldM. This manifold for

the STU model is SLð2;RÞ=Uð1Þ�3. In the following wewill denote the three complex scalar fields as

za � xa � iya; ya > 0; a ¼ 1; 2; 3: (2)

With these definitions the metric and the connection on thescalar manifold are

Ga �b ¼ �a �b

ð2yaÞ2 ; �aaa ¼ �i

ya: (3)

The metric above can be derived from the Kahler potential

K ¼ � logð8y1y2y3Þ (4)

as Ga �b ¼ @a@ �bK. For the STU model the scalar dependentvector couplings ReN IJ and ImN IJ take the followingform:

ReN IJ ¼2x1x2x3 �x2x3 �x1x3 �x1x2

�x2x3 0 x3 x2

�x1x3 x3 0 x1

�x1x2 x2 x1 0

0BBB@

1CCCA; (5)

ImN IJ ¼ �y1y2y3

1þ�x1

y1

�2 þ

�x2

y2

�2 þ

�x3

y3

�2 � x1

ðy1Þ2 � x2

ðy2Þ2 � x3

ðy3Þ2

� x1

ðy1Þ21

ðy1Þ2 0 0

� x2

ðy2Þ2 0 1ðy2Þ2 0

� x3

ðy3Þ2 0 0 1ðy3Þ2

0BBBBBBB@

1CCCCCCCA: (6)

We note that these vector couplings can be derived fromthe holomorphic prepotential

FðXÞ ¼ X1X2X3

X0; XI ¼ ðX0; X0zaÞ; (7)

via the standard procedure characterizing special Kahlergeometry [39].

For the physical motivation of Eq. (1) we note that whentype IIA string theory is compactified on a T6 of the formT2 � T2 � T2 one recovers N ¼ 8 supergravity in d ¼ 4with 28 vectors and 70 scalars taking values in the sym-metric space E7ð7Þ=SUð8Þ. This N ¼ 8 model with an on

shell U-duality symmetry E7ð7Þ has a consistent N ¼ 2truncation with 4 vectors and three complex scalars whichis just the STU model [24,25]. The D0�D2�D4�D6branes wrapping the various T2 give rise to four electricand four magnetic charges defined as

PI ¼ 1

4�

ZS2F I; QI ¼ 1

4�

ZS2GI; I ¼ 0;1;2;3;

(8)

where

G I ¼ �N IJFþI; F�I�� ¼ F I

�� i

2"��F I��: (9)

These charges can be organized into symplectic pairs

� � ðPI;QJÞ (10)

and have units of length. They are related to the dimen-sionless quantized charges by some dressing factors. Usingthe variable � ¼ 1=r ¼ 1=jxj and normalizing the asymp-totic moduli as

yað0Þ ¼ 1; xað0Þ ¼ Ba; (11)

the dressing factors are essentially the masses of the under-lying branes [31].We are interested in static, spherically symmetric, ex-

tremal black hole solutions associated to the (1) action. Theansatz for the metric is

ds2 ¼ �e2Uð�Þdt2 þ e�2Uð�Þdx2; (12)

where the warp factor is a function of � ¼ 1=r. Putting thisansatz into (1) we obtain a one-dimensional effectiveLagrangian for the radial evolution of the quantitiesUð�Þ, zað�Þ, as well as the electric and magnetic potentials[40]

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L ðUð�Þ; zað�Þ; �z �að�ÞÞ ¼�dU

d�

�2 þGa �a

dza

d�

d�z �a

d�

þ e2UVBHðz; �z; P;QÞ; (13)

and the constraint�dU

d�

�2 þGa �a

dza

d�

d�z �a

d�� e2UVBHðz; �z; P;QÞ ¼ 0: (14)

Here our quantity of central importance is the black holepotential VBH which is depending on the moduli as well ason the charges. Its explicit form is given by

VBH ¼ 1

2PI QI

� � ð�þ ��1�ÞIJ �ð��1ÞJI�ð��1�ÞIJ ð��1ÞIJ

� �PJ

QJ

� �;

(15)

where the matrices � ¼ ReN and� ¼ ImN are the onesof Eqs. (5) and (6). The explicit form of ��1 is

��1 ¼ �1

y1y2y3

1 x1 x2 x3

x1 jz1j2 x1x2 x1x3

x2 x1x2 jz2j2 x2x3

x3 x1x3 x2x3 jz3j2

0BBB@

1CCCA: (16)

An alternative expression for VBH can be given in terms ofthe central charge of N ¼ 2 supergravity, i.e. the charge ofthe graviphoton.

VBH ¼ Z �ZþGa �bðDaZÞð �D �b�ZÞ; (17)

where for the STU model

Z ¼ eK=2W

¼ eK=2ðQ0 þ z1Q1 þ z2Q2 þ z3Q3 þ z1z2z3P0

� z2z3P1 � z1z3P2 � z1z2P3Þ; (18)

and Da is the Kahler covariant derivative

DaZ ¼ ð@a þ 12@aKÞZ; (19)

and W is the superpotential.Extremization of the effective Lagrangian Eq. (13) with

respect to the warp factor and the scalar fields yields theEuler-Lagrange equations

€U ¼ e2UVBH; €za þ �abc _z

b _zc ¼ e2U@aVBH: (20)

In these equations the dots denote derivatives with respectto �. These equations taken together with the constraintEq. (14) determine the black hole solutions whose quantuminformation theoretic interpretation we are interested in.

III. THREE-QUBIT STATES

It is useful to reorganize the charges of the STU modelinto the 8 amplitudes of a three-qubit state

j�i ¼ Xl;k;j¼0;1

�lkjjlkji; jlkji � jli3 � jki2 � jji1;

(21)

where

1ffiffiffi2

p P0; P1; P2; P3

�Q0; Q1; Q2; Q3

� �

¼ �000; �001; �010; �100

�111; �110; �101; �011

� �: (22)

Notice that we have introduced the convention of labelingthe qubits from right to left. Moreover, for convenience wehave also included a factor 1ffiffi

2p into our definition. The state

j�i is a three-qubit state of a very special kind. First of allthis state defined by the charges need not have to benormalized. Moreover, the amplitudes of this state arenot complex numbers but real ones. As a next step wecan define a new three-qubit state j�i depending on thecharges � and also on the moduli [3,7]. This new state willbe a three-qubit state with 8 complex amplitudes. However,as we will see it is really a real three-qubit state, since it isUð2Þ�3 equivalent to a one with 8 real amplitudes.In order to motivate our definition of the new state j�i

we notice that [7]

VBH ¼ 1

y1y2y3h�j jz3j2 �z3

�z3 1

� �� jz2j2 �z2

�z2 1

� �

� jz1j2 �z1

�z1 1

� �j�i: (23)

Now we define the state j�i as

j�ðza; �z �a;�Þi ¼ eK=2 �z3 �1�z3 1

� �� z2 �1

�z2 1

� �

� �z1 �1�z1 1

� �j�i: (24)

Introducing the matrices

S a � 1ffiffiffiffiffiffiffiffi2ya

p �za �1�za 1

� �¼ USa

� 1ffiffiffi2

p i �1i 1

� �1ffiffiffiffiffiya

p ya 0�xa 1

� �;

a ¼ 1; 2; 3:

(25)

With this notation we have

j�i ¼ ðS3 � S2 � S1Þj�i¼ ðU �U �UÞðS3 � S2 � S1Þj�i: (26)

This means that the state j�i up to a phase for all values ofthe moduli is in the GLð2;CÞ�3 orbit of the charge statej�i. Obviously the state j�i is an unnormalized three-qubitone with 8 complex amplitudes. However, it is not agenuine complex three-qubit state but rather one which isUð2Þ�3 equivalent to a real one. This should not come as a


026002-4

surprise since the symmetry group associated with the STUmodel is not GLð2;CÞ�3 but rather SLð2;RÞ�3.

Using these definitions we can write the black holepotential in the following nice form:

VBH ¼ k�k2: (27)

Here the norm is defined using the usual scalar product inC8 ’ C2 � C2 � C2 with complex conjugation in the firstfactor. Since the norm is invariant under Uð2Þ�3 our choiceof the first unitary matrix of Eq. (25) is not relevant in thestructure of VBH. We could have defined a new modulidependent real state instead of the complex one j�i byusing merely the SLð2;RÞ matrices of Eq. (25) for theirdefinition. However, we prefer the complex form ofEq. (26) since it will be useful later.

It is instructive to write out explicitly the amplitudes ofour complex three-qubit state j�i:ffiffiffi

2p

�000 ¼ eK=2Wð�z3; �z2; �z1Þ;ffiffiffi2

p�111 ¼ �eK=2Wðz3; z2; z1Þ;

(28)

ffiffiffi2

p�110 ¼ eK=2Wðz3; z2; �z1Þ;ffiffiffi

2p

�001 ¼ �eK=2Wð�z3; �z2; z1Þ;(29)

with the remaining amplitudes arising by cyclic permuta-tion. Notice also that we have the property (reality condi-tion)

�000 ¼ � ��111; �110 ¼ � ��001;

�101 ¼ � ��010; �011 ¼ � ��100;(30)

which can be written as

j ��i þ ð�1 � �1 � �1Þj�i ¼ 0 (31)

via the bit-flip operator �1. Using this in Eq. (27) we canwrite VBH in the alternative form [38]

VBH ¼ eKðjWðz3; z2; z1Þj2 þ jWðz3; z2; �z1Þj2þ jWðz3; �z2; z1Þj2 þ jWð�z3; z2; z1Þj2Þ: (32)

As a motivation for our particular definition for j�i wenotice that ffiffiffi

2p

�000 ¼ �Z; (33)

i.e. the amplitudes �111 and�000 are related to the centralcharge and its complex conjugate. The remaining ampli-tudes are simply arising by conjugating one or two moduli.Moreover, one can show [7] that the flat covariant deriva-tives with respect to the moduli act on our j�i as bit-fliperrors. Explicitly we have

D1j�i ¼ ðI � I � �þÞj�i;D �1

j�i ¼ ðI � I � ��Þj�i;D2j�i ¼ ðI � �þ � IÞj�i;D �2

j�i ¼ ðI � �� IÞj�i;D3j�i ¼ ð�þ � I � IÞj�i;D �3

j�i ¼ ð�� I � IÞj�i:

(34)

Here the operators �� act as

�þj0i¼ j1i; �þj1i¼ 0; ��j0i¼ 0; ��j1i¼ j0i;(35)

and the flat covariant derivatives are defined as D1 ¼�2iy1D1, D �1¼ 2iy1D�1, etc. where

D1Wðz3; z2; z1Þ ¼ Wðz3; z2; �z1Þ�z1 � z1

;

D�1Wðz3; z2; z1Þ ¼ 0; etc:

(36)

Hence the flat covariant derivatives are acting on our three-qubit state j�i as the operators of projective errors knownfrom the theory of quantum error correction. Alternativelyone can look at the action of the combination Da þD �a

ðD1 þD �1Þj�i ¼ ðI � I � �1Þj�i; etc: (37)

According to our previous paper [7] it is illuminating touse the discrete Fourier transform of our three-qubit statej�i. At the event horizon the moduli are stabilized due tothe attractor mechanism, and their stabilized values can beexpressed in terms of the charges. These stabilized valuesgive rise to entangled states on the horizon of a very specialform. It was shown [3,7] that for BPS solutions these statesare of a generalized GHZ form [27], and for the simplenon-BPS solutions [38] they are graph states [28] wellknown from quantum information theory.The discrete Fourier (Hadamard) transformation is im-

plemented by acting on j�i by H �H �H where

H ¼ 1ffiffiffi2

p 1 11 �1

� �: (38)

Hence the Fourier transformed basis states are defined as

j~0i � 1ffiffiffi2

p ðj0i þ j1iÞ ¼ Hj0i;

j~1i � 1ffiffiffi2

p ðj0i � j1iÞ ¼ Hj1i:(39)

Since H�1H ¼ �3 and H�3H ¼ �1, the operator �1 isacting on the Hadamard transformed base as a phase (sign)flip operator and vice versa. The important corollary of thisobservation is that in the theory of quantum error correc-tion once we have found a means for correcting bit-fliperrors using a discrete Fourier transform the same tech-nique can be used for correcting phase flip ones. It was


026002-5

shown [7] that the phase flip errors in the discrete Fouriertransformed base correspond to flipping the sign of certaincharges of the symplectic vector �. For BPS solutions thesephase flips are suppressed, and for the simple non-BPSsolutions only errors of a very special kind are allowed.

Now we would like to gain some more insight into theseinteresting results by studying the solutions even awayfrom the horizon. For this the full solution of the flow inmoduli space is needed. For BPS solutions we can use thewell-known results [29,41] and for the non-BPS solutionsthe recently found seed solutions [30–32] and the mostgeneral non-BPS flows [11] generalizing the simple non-BPS solutions [38,42,43].

Our main calculational tool will be to consider thediscrete Fourier transform of our j�i

j ~�i ¼ ðH �H �HÞj�i¼ ðP � P � P ÞðS3 � S2 � S1Þj�i; (40)

where

P ¼ i 00 �1

� �(41)

is just i times the usual phase gate known from quantuminformation theory. Hence from Eq. (40) we see that theFourier transformed state is up to some important complexphase factors lying on the SLð2;RÞ�3 orbit of the chargestate j�i. Now solving Eqs. (20) with the constraint ofEq. (14) yields the flow zað�Þ on moduli space. To this

flow we can associate a corresponding one j ~�ð�Þi of theFourier transformed state. Our aim for the following sec-

tions is to look at the structure of j ~�ð�Þi for BPS and thenon-BPS seed solutions.

IV. BPS SOLUTIONS

In this section we would like to study the behavior ofBPS solutions in the three-qubit picture. In particular wewould like to see how the three-qubit state of Eq. (24)behaves as a function of �, answering to the flow zað�Þ inmoduli space. This three-qubit picture is natural as the U-duality group is SLð2;RÞ�3 which is a subgroup ofSpð8;RÞ; hence we expect that the usual symplectic invar-iants occurring in the formalism of the STU model shouldboil down to the corresponding SLð2;RÞ�3 i.e. three-qubitones.

As the first step in order to present the solutions capableof incorporating a wider range of asymptotic data withB fields we define a set of harmonic functions as

H ð�Þ ¼ ��þ ��; i:e: HI ¼ �PI þ PI�;

HJ ¼ �QJ þQJ; � I; J ¼ 0; 1; 2; 3:(42)

We can alternatively encode the asymptotic data [H ð0Þ ¼��] into a three-qubit state

j ��i ¼ Xl;k;j¼0;1

��lkjjlkji;

��000; ��001; ��010; ��100

��111; ��110; ��101; ��011

!� 1ffiffiffi

2p

�P0; �P1; �P2; �P3

� �Q0; �Q1; �Q2; �Q3

!;

(43)

which plays a role similar to the charge state of Eq. (21).

Notice that unlike for �� in Eq. (42) in the state j ��i we alsoincluded a factor of 1ffiffi

2p for convenience.

For later use we also define a � dependent three-qubitstate as

jH ð�Þi � j ��i þ �j�i: (44)

This state in the limits � ! 0 and � ! 1 characterizes theasymptotic data and the charge configuration respectively.Let us now define Cayley’s hyperdeterminant [44,45]

Dðjc iÞ for an arbitrary three-qubit state jc i ¼Plkj¼0;1c lkjjlkji with amplitudes

c 000; c 001; c 010; c 100

c 111; c 110; c 101; c 011

� �

� c 0; c 1; c 2; c 4

c 7; c 6; c 5; c 3

� �(45)

as

Dðjc iÞ � ðc 0c 7Þ2 þ ðc 1c 6Þ2 þ ðc 2c 5Þ2 þ ðc 4c 3Þ2� 2ðc 0c 7Þ½ðc 1c 6Þ þ ðc 2c 5Þ þ ðc 4c 3Þ� 2½ðc 1c 6Þðc 2c 5Þ þ ðc 1c 6Þðc 4c 3Þþ ðc 2c 5Þðc 4c 3Þ þ 4c 0c 1c 2c 4

þ 4c 7c 6c 5c 3: (46)

Dðjc iÞ is permutation and SLð2;CÞ�3 invariant and underthe group GLð2;CÞ�3 transforms as

Dðc Þ � ðdetG3Þ2ðdetG2Þ2ðdetG1Þ2Dðc Þ;G3 �G2 �G1 2 GLð2;CÞ�3: (47)

Notice that for a general charge state j�i such as the one ofEq. (21) we have

� 4Dðj�iÞ ¼ I4ð�Þ; (48)

where I4ð�Þ is the usual quartic invariant known fromstudies concerning the STU model

I4ð�Þ ¼ �ðPIQIÞ2 þ 4½ðP1Q1ÞðP2Q2Þ þ ðP2Q2ÞðP3Q3Þþ ðP1Q1ÞðP3Q3Þ þ 4Q0P

1P2P3 � 4P0Q1Q2Q3:

(49)

According to the general theory the data giving rise toBPS black hole solutions are incorporated into a H ð�Þsubject to two constraints [29,41]


026002-6

I4ð ��Þ ¼ �4Dðj ��iÞ ¼ 1; ð�; ��Þ � PI �QI �QJ�PJ ¼ 0:

(50)

As far as these constraints hold we can completely char-acterize any I4ð�Þ> 0 solution by generalizing the attrac-tor equations to the so-called stabilization equations [29].The warp factor of Eq. (12) is

e�4Uð�Þ ¼ I4ðH ð�ÞÞ ¼ �4DðjH ð�ÞiÞ: (51)

Since for normalized states the quantity

0 �123 � 4jDðjc iÞj 1 (52)

is characterizing the tripartite entanglement of three-qubitsystems [45] we observe that the warp factor can beregarded as a � ¼ 1

r dependent entanglement measure de-

scribing the tripartite entanglement of a state encapsulatingthe details of the charge configuration and the asymptoticvalues for the moduli, i.e.

e�4UðrÞ ¼ �123ðjH ðrÞiÞ: (53)

In this picture the first constraint of Eq. (50) means that forthe state of Eq. (44) the asymptotic value of the ‘‘three-tangle’’ is normalized to 1.

In order to find an entanglement based meaning for thesecond constraint of Eq. (50) notice that the 16 quantities

of the states j�i and j ��i can be organized into a four-qubitstate. Indeed let us define a state j�iwith its 16 amplitudes�mlkj given by

�0lkj � �lkj; �1lkj � ��lkj: (54)

Then a four-qubit entanglement measure invariant underSLð2;CÞ�4 is given by [46] j�1234j, where�1234ðj�iÞ � �0�15 � �1�14 � �2�13 þ �3�12 � �4�11

þ �5�10 þ �6�9 � �7�8; (55)

where for simplicity we again used the decimal labeling.Now using the definitions as given by Eqs. (21) and (43) itis straightforward to check that

ð�; ��Þ ¼ �1234ðj�iÞ; (56)

meaning that our second constraint is equivalent to thevanishing of the entanglement invariant �1234. Hence weconclude that both of the constraints describing the BPSblack hole solutions have a characteristic meaning in ourentanglement based reformulation. Recalling also that thevalue of �123ðj�iÞ is related to the entropy of the BPS STUblack hole [2,3,24] we can summarize these results as

S ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�123ðj�iÞ

q; �123ðj ��iÞ ¼ 1; �1234ðj�iÞ ¼ 0:

(57)

Now in order to set the stage for the following general-izations we review and slightly extend the known resultsconcerning the distillation procedure for BPS solutions [3].

In the following for simplicity we consider the D0�D4system. In this case we have Q0 > 0 and Pi > 0 but P0 ¼Qi ¼ 0. Generalizing the simple BPS solution we alsoinclude nontrivial B fields as follows [31]. First we define

our state j ��i with amplitudes for Eq. (43) as

�P 0 ¼ 1ffiffiffi2

p sin�; �P1;2;3 ¼ 1ffiffiffi2

p ðcos�þ sin�B1;2;3Þ;(58)

�Q 1 ¼ 1ffiffiffi2

p ðsin�½1� B2B3 � cos�½B1 þ B2Þ;

and cyclic permutations;

(59)

�Q0 ¼ 1ffiffiffi2

p ½ðB1 þ B2 þ B3 � B1B2B3Þ sin�

þ ð1� B1B2 � B2B3 � B1B3Þ cos�: (60)

Here B1, B2, and B3 are related to the asymptotic values forthe moduli as

zaj�¼0 ! Ba � i; (61)

i.e. the asymptotic volume moduli are normalized, but wekeep the asymptotic B fields as free variables.In terms of our three-qubit states there is a nice way of

understanding this choice for �PI and �QI and also themeaning of the additional parameter �. First just like in

Eq. (26) we define a new moduli dependent state j ��ð�Þi asj ��ð�Þi � ðS3ð�Þ � S2ð�Þ � S1ð�ÞÞj ��i; (62)

where for the definition of S1;2;3, see Eq. (25). We would

like to see how this state behaves at the asymptoticallyMinkowski region. A straightforward calculation usingEq. (61) shows that

j ��ð0Þi ¼ 1ffiffiffi2

p ðe�i�j000i � ei�j111iÞ; (63)

i.e. the parameter � is related to the phase of a generalizedGHZ state. Notice that thanks to our inclusion of the factor1ffiffi2

p in the (43) definition of j ��i this state is normalized. On

the other hand, according to Eq. (27) a similar inclusion forthe definition of the companion state j�i [Eq. (22)] has alsofixed the norm of the state j�i to be the black holepotential. Knowing that for the normalized generalizedGHZ state of Eq. (63) �123 ¼ 4jDj ¼ 1 by virtue of

Eqs. (47), (49), and (62) we immediately get I4ð ��Þ ¼ 1,i.e. the first of our constraint is satisfied.In our recent paper [5] it was shown that the attractor

mechanism for STU BPS black holes can be reinterpretedas a distillation mechanism of a GHZ state at the horizon(� ¼ 1) of the form

j�ð1Þi ¼ ðI4ð�ÞÞ1=4 1ffiffiffi2

p ðe�i�j000i � ei�j111iÞ; (64)


026002-7

where the explicit expression of � is given by [7]

cot� ¼ 1

P0

@

@Q0

ffiffiffiffiffiffiffiffiffiffiffiI4ð�Þ

q; (65)

where for the definition of I4ð�Þ, see Eq. (49). ComparingEqs. (63) and (64) we see that we have an attractor at thehorizon (� ¼ 1) and an attractor at the asymptoticallyMinkowski region (� ¼ 0). These attractors can be de-scribed by the flows of the charge and moduli dependent

states j�ð�Þi and j ��ð�Þi, respectively, each of them pro-ducing a maximally entangled GHZ state at the attractorpoints defined by � ¼ 1 and � ¼ 0. The attractor at � ¼ 0giving rise to the GHZ state of Eq. (63) is the well-knownattractor at infinity [36], a map between the 6 real moduliand the 8 constants in the harmonic functions subject to 2constraints. We can summarize these considerations bynoting that for the relevant flows and attractor points wehave

k ��ð0Þk2 ¼ 1; k�ð1Þk2 ¼ VBHð1Þ ¼ffiffiffiffiffiffiffiffiffiffiffiI4ð�Þ

q: (66)

Now we compare the flows j ��ð�Þi and j�ð�Þi by calcu-lating

h ��ð�Þj�3 � �3 � �3j�ð�Þi: (67)

This quantity is the transition amplitude between thephase-flipped state ð�3 � �3 � �3Þj�ð�Þi and the one

j ��ð�Þi. Using Eqs. (26) and (62) and the fact thatUy�3U ¼ ��2 and ST�2S ¼ �2 for S 2 SLð2;RÞ, weget

h ��ð�Þj�3 � �3 � �3j�ð�Þi ¼ �h ��j�2 � �2 � �2j�i¼ ið ��;�Þ ¼ i�1234ðj�iÞ ¼ 0:

(68)

This shows that the phase (sign)-flipped version of �ð�Þ isalways orthogonal to the companion state ��ð�Þ. It is alsoclear that this amplitude is purely topological in origin(e.g., in the type IIB duality frame it is related to theintersection product [41] on T6). Because of the fact thatH�3H ¼ �1, where �1 is the bit-flip operator we canalternatively conclude that the bit flipped version of theFourier transform of one of the states is orthogonal to theFourier transform of the other. Recall also that according toEq. (37) such bit-flip errors are related to the action of theflat covariant derivatives with respect to the moduli.

As a last application of the vanishing of the amplitude ofEq. (68), Eqs. (33) and (63) give the meaning of theparameter � as the phase of the central charge Z (up to ashift by �). These results shed some light on the quantuminformation theoretic meaning of the second constraint ofEq. (50).

In closing this section we present the well-known solu-tion of the stabilization equations giving the moduli fieldsas a function of � [29,31,41]

z1 ¼ �H1H1 þH0H

0 þH2H2 þH3H

3 � ie�2U

2ðH2H3 �H0H1Þ; (69)

with the remaining equations for z2 and z3 arising aftercyclic permutation of the numbers 1, 2, and 3. Here HI ¼�PI þ PI�, HJ ¼ �QJ þQJ�, with I, J ¼ 0, 1, 2, 3, and thewarp factor is given by Eq. (51). It is straightforward tocheck that the solution zað�Þ satisfies the correct asymp-totic behavior of Eq. (61) and

zað1Þ ¼ �i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Q0P

a

sabcPbPc

s; sabc � j"abcj: (70)

By virtue of Eqs. (28)–(30) and (61) it is clear that theasymptotic values of the 8 amplitudes of j�ð0Þi are gen-erally nonzero. However, the flow zað�Þ in moduli spacegiving rise to the flow j�ð�Þi results in a GHZ state at thehorizon having merely 2 nonvanishing amplitudes [seeEq. (64)].This distillation process is the one that was studied for

BPS solutions [3]. Later a non-BPS generalization was alsogiven [7]. In this case the distillation process gives rise tograph states [28] at the horizon. However, the analysis inthese papers was restricted merely to double extremalsolutions [25] for which the moduli are constant evenaway from the horizon. Hence in these studies the impor-tant question of how the distillation process becomes un-folded as � changes have not been addressed. Our aim inthe next section is to investigate such issues by studying thenon-BPS flow explicitly.

V. THE FLOW FOR THE NON-BPS D0�D4SYSTEM

In this section we study the flow j�ð�Þi answering thefull radial flow zað�Þ obtained for the 5 parameter family ofnon-BPS seed solutions of aD0�D4 system described byGimon et al. [31]. More precisely it turns out to be reward-ing to study the properties of a related flow jð�Þi obtainedby multiplying j�ð�Þi by the warp factor. Then we showthat the Fourier amplitudes of this new state satisfy a firstorder system of differential equations in accordance withour expectation coming from previous studies [33,34,47].First we address the quantum information theoretic as-

pects of the seed solution. For the D0�D4 system wechose Q0 < 0 and Pa > 0, a ¼ 1, 2, 3. Let us define

za ¼ xa � iya; xa ¼ Rata;

ya ¼ Raea ; Ra ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2Q0P

a

sabcPbPc

s:

(71)

From Eq. (20) the equations to be solved are

d

d�ð _tae�2aÞ ¼ 2e2U

@VBH

@ta; a ¼ 1; 2; 3; (72)


026002-8

€a þ ð _tae�aÞ2 ¼ 2e2U@VBH

@a

; a ¼ 1; 2; 3; (73)

€U ¼ e2UVBH: (74)

Moreover, according to Eq. (14) we also have the con-straint

_U 2 þ 1

4

Xa

ð _2a þ ½ _tae�a2Þ ¼ e2UVBH: (75)

As the first step we define a new three-qubit state byincorporating also the warp factor as

jð�Þi � eUð�Þj�ð�Þi ¼ eUð�ÞðS3 � S2 � S1Þj�i: (76)

This state depends on the charges, the moduli and the warpfactor. The discrete Fourier transform of this state [for thedefinitions see Eqs. (25), (38), (40), and (41)]

j~ð�Þi ¼ ðH �H �HÞjð�Þi¼ eUð�ÞðP � P � P ÞðS3 � S2 � S1Þj�i; (77)

will play a particularly important role in the following.For later use we write out explicitly the amplitudes of

this Fourier transformed state j~ð�Þi~001 ¼ 1

2jI4j1=4e�þ2þ3 ; ~010 ¼ 12jI4j1=4e�þ1þ3 ;

~100 ¼ 12jI4j1=4e�þ1þ2 ; (78)

i~110 ¼ 12jI4j1=4e�þ1ðt2 þ t3Þ;

i~101 ¼ 12jI4j1=4e�þ2ðt1 þ t3Þ;

i~011 ¼ 12jI4j1=4e�þ3ðt1 þ t2Þ;

(79)

~ 000 ¼ 0;

~111 ¼ �12jI4j1=4e�ð1þ t1t2 þ t2t3 þ t3t1Þ:

(80)

Here

� ¼ U� 12ð1 þ2 þ2Þ;

jI4j ¼ �I4 ¼ �4Q0P1P2P3 > 0:

(81)

Now in terms of these amplitudes the equations to besolved can be written as�

d

d�� _a

�ð _tae�aÞ ¼ �2h~jYaj~i; (82)

€a þ ð _tae�aÞ2 ¼ 2h~jZaj~i; (83)

€U ¼ h~j~i; (84)

where the operators Z1 � I � I � �3, Z2 � I � �3 � I,and Z3 ¼ �3 � I � I describe phase flips of the ath qubitin the Fourier transformed base. The quantities Ya aredefined using �2 accordingly. Notice also that the opera-

tors 12 ð1þZaÞ, where 1 is the 8� 8 unit matrix, are

projection operators. Hence by adding the one-half ofEq. (83) to Eq. (84) gives rise to three equations whichcontain merely three amplitudes (~0 ¼ 0) on the right-hand side. These manipulations also justify the introduc-tion of a new variable [30,31]

�a � Uþ 12a: (85)

Using this new variable instead of Eq. (83) we can use theones

€� 1 � 2

�i

2_tae

�a

�2 ¼ 2ð~2

2 þ ~24 � ~2

6Þ; (86)

€� 2 � 2

�i

2_t2e

�2

�2 ¼ 2ð~2

1 þ ~24 � ~2

5Þ; (87)

€� 3 � 2

�i

2_t3e

�3

�2 ¼ 2ð~2

1 þ ~22 � ~2

3Þ; (88)

where from now on we use decimal labeling for ouramplitudes. For Eq. (82) we have the form

dd� � _1 0 0

0 dd� � _2 0

0 0 dd� � _3

0B@

1CA

i2_t1e

�1

i2_t2e

�2

i2_t3e

�3

0B@

1CA

¼ 2~7 �~4 �~2

�~4 ~7 �~1

�~2 �~1 ~7

0@

1A ~6

~5

~3

0@

1A; (89)

and the sum of Eq. (84) and � 12 the sum of Eq. (83) gives

€�þ 2Xa

�i

2_tae

�a

�2 ¼ 4~2

7 � 2ð~26 þ ~2

5 þ ~23Þ: (90)

Now, the crucial observation which enables an explicitconstruction of the seed solutions is the fact that in theFourier transformed basis we have seven nonvanishingamplitudes, and the constraint Eq. (75) is consisting ofsquares of seven terms. Using the decimal labeling forthe amplitudes of j~i this constraint can be written in theform

_� 2 þ Xa�b�c

ð _�a þ _�b � _�cÞ2 þXa

ðe�a _taÞ2

¼ 4ð27 þ 2

1 þ 22 þ 2

4 � 26 � 2

5 � 23Þ: (91)

A natural choice to satisfy this constraint up to a sign is

� ~7 ¼ 1

2_�; �~6 ¼ i

2e�1 _t1;

� ~5 ¼ i

2e�2 _t2; �~3 ¼ i

2e�3 _t3;

(92)

�~1 ¼ 12ð _�1 � _�2 � _�3Þ; �~2 ¼ 1

2ð _�2 � _�1 � _�3Þ;� ~4 ¼ 1

2ð _�3 � _�1 � _�2Þ: (93)


026002-9

Comparing these with the explicit form of the Fourieramplitudes of Eqs. (78)–(80) with ~1;2;4 rewritten as

~1 ¼ 12jI4j1=4e�2þ�3��1 ; ~2 ¼ 1

2jI4j1=4e�3þ�1��2 ;

~4 ¼ 12jI4j1=4e�1þ�2��3 ; (94)

we get the following set of first order differential equa-tions:

_t a ¼ �jI4j1=4ðtb þ tcÞe3�a��b��c ; (95)

_� ¼ �jI4j1=4e�ð1þ t1t2 þ t2t3 þ t3t1Þ; (96)

ð _�a þ _�b � _�cÞ ¼ �jI4j1=4e�aþ�b��c ; (97)

where a � b � c. The solutions to these equations withthe upper sign choice were given by Gimon et al. [31].Before recalling these solutions we show that these choicesautomatically solve Eqs. (86)–(90). Let us substitute in-stead of i

2_tae

�a occurring in these equations the Fourier

amplitudes of Eqs. (92). Then we get

€� ¼ 4ð~27 � ~2

6 � ~25 � ~2

3Þ; (98)

€� 1¼2ð~22þ ~2

4Þ; €�2¼2ð~21þ ~2

4Þ; €�3¼2ð~21þ ~2

2Þ:(99)

In Eq. (89) we also replace ~7 by12_� (we have chosen the

upper sign) to get

d

d�

~6

~5

~3

0BB@

1CCA ¼ 2

12 ð _�þ _1Þ �~4 �~2

�~412 ð _�þ _2Þ �~1

�~2 �~112 ð _�þ _3Þ

0BBB@

1CCCA

�~6

~5

~3

0BB@

1CCA: (100)

Now using the explicit expressions for the Fourier ampli-tudes it is easy to check that these equations are indeedsatisfied.

An interesting possibility is to write down these equa-tions as first order equations for the Fourier amplitudes.From Eqs. (98) and (100) we get

d

d�

~7

~6

~5

~3

0BBBBB@

1CCCCCA¼ 2

12_� �~6 �~5 �~3

0 12 ð _�þ _1Þ �~4 �~2

0 �~412 ð _�þ _2Þ �~1

0 �~2 �~112 ð _�þ _3Þ

0BBBBBB@

1CCCCCCA

�

~7

~6

~5

~3

0BBBBB@

1CCCCCA: (101)

Similarly from Eq. (99) using Eq. (94) the correspondingequation is

d

d�

~1

~2

~4

0@

1A ¼ 2

�~1 0 00 �~2 00 0 �~4

0@

1A ~1

~2

~4

0@

1A: (102)

An alternative form for Eq. (101) can be given by noticingthat

_�þ _a ¼ 2 _U� ð _�b þ _�b � _�aÞ¼ 2 _Uþ jI4j1=4e�bþ�c��a ¼ 2 _Uþ 2~1;2;4;

(103)

hence we have

d

d�

~7

~6

~5

~3

0BBBBB@

1CCCCCA ¼ 2

~7 �~6 �~5 �~3

0 _Uþ ~1 �~4 �~2

0 �~4_Uþ ~2 �~1

0 �~2 �~1_Uþ ~4

0BBBBB@

1CCCCCA

�

~7

~6

~5

~3

0BBBBB@

1CCCCCA: (104)

Equations (102) and (104) are first order equations con-taining the Fourier amplitudes of our charge, moduli andwarp factor dependent states and _U. However, we canexpress _U in terms of some of the ~’s as follows. Let usdefine

w � 12ð~7 � ~1 � ~2 � ~4Þ: (105)

Then from Eqs. (102) and (104) we get

d

d�w ¼ ð~1Þ2 þ ð~2Þ2 þ ð~4Þ2 þ ð~7Þ2

� ð~6Þ2 � ð~5Þ2 � ð~3Þ2 ¼ e2UVBH: (106)

Hence fom Eq. (84)

d

d�U ¼ w � eUW : (107)

The new quantity W

W ¼ 12ð ~�7 � ~�1 � ~�2 � ~�4Þ

¼ 12ð ~�111 � ~�001 � ~�010 � ~�100Þ; (108)

is the fake superpotential [11,33] where it is easy to checkthat its explicit form coincides with the negative of the oneas given by Eq. (6.8) of the paper by Bellucci et al. [11].Notice also that the fake superpotential contains onlyFourier amplitudes of odd parity of the charge and modulidependent three-qubit state j�i. This will be of someimportance in the next section.


026002-10

Since we have _U ¼ 12 ð~7 � ~1 � ~2 � ~4Þ ¼ wwe can

write Eq. (104) in the final form

d

d�

~7

~6

~5

~3

0BBBBB@

1CCCCCA ¼ 2

~7 �~6 �~5 �~3

0 wþ ~1 �~4 �~2

0 �~4 wþ ~2 �~1

0 �~2 �~1 wþ ~4

0BBBBB@

1CCCCCA

�

~7

~6

~5

~3

0BBBBB@

1CCCCCA: (109)

Hence Eqs. (102) and (109) show that in the case of theseed solution for the non-BPS Z � 0 D0�D4 system the� derivatives of the Fourier amplitudes of our charge,moduli, and warp factor-dependent state ji can be ex-pressed entirely in terms of the Fourier amplitudes.

VI. THE ATTRACTOR MECHANISM AS ADISTILLATION PROCEDURE

In this section we would like to demonstrate how theradial flow studied in the previous section gives rise to thedistillation of a special three-qubit state at the black holehorizon. In order to see this procedure unfolding all wehave to do is to use the solutions of the first order equations(95)–(97) to obtain explicit expressions for the Fourieramplitudes ~lkjð�Þ. It means that starting from the asymp-

totic values ~lkjð0Þ in the region with Minkowski geometry

at the limit � ! 1 we obtain the ones ~lkjð1Þ at the

horizon with adS2 � S2 geometry. The solutions ofEqs. (95)–(97) are [31]

e�aþ�b��c ¼ 1

dc þ jI4j1=4�� 1

hc; a � b � c;

a; b; c ¼ 1; 2; 3;

(110)

ta ¼ B

hbhc; e�� ¼ �h0 � B2

h1h2h3;

h0 � �d0 � jI4j1=4�;(111)

where

da ¼ jI4j1=4ffiffiffi2

pPa

; d0 ¼ �jI4j1=4ffiffiffi2

pQ0

ð1þ B2Þ; (112)

where for the definition of jI4j, see Eq. (81). Notice thataccording to Eqs. (11) and (71) for this 5 parameter solu-tion we have

za ¼ Ra

B� ie�2U

12 sabchbhc

; e�4U ¼ �h0h1h2h3 � B2;

(113)

hence B � x1ð0Þ ¼ x2ð0Þ ¼ x3ð0Þ.In order to write down the explicit form of the ampli-

tudes of our charge, moduli, and warp factor dependentFourier transformed state it is useful to introduce the newharmonic functions

Hað�Þ ¼ 1ffiffiffi2

p þ Pa� ¼ Pa

jI4j1=4hað�Þ;

H0ð�Þ ¼ � 1ffiffiffi2

p ð1þ B2Þ þQ0� ¼ � Q0

jI4j1=4h0;

(114)

and the warp factor

e�4Uð�Þ ¼ �4H0ð�ÞH1ð�ÞH2ð�ÞH3ð�Þ � B2: (115)

Using these results we get the following results for theFourier amplitudes for our charge, moduli, and warp factordependent state j~i

~ 000ð�Þ ¼ 0; ~001ð�Þ ¼ P1

2H1ð�Þ ;

~010ð�Þ ¼ P2

2H2ð�Þ ; ~100ð�Þ ¼ P3

2H3ð�Þ ;(116)

~ 110ð�Þ ¼ � i

2e2Uð�Þ

�P2

H2ð�Þ �P3

H3ð�Þ; (117)

~ 101ð�Þ ¼ � i

2e2Uð�Þ

�P1

H1ð�Þ �P3

H3ð�Þ; (118)

~ 011ð�Þ ¼ � i

2e2Uð�Þ

�P1

H1ð�Þ �P2

H2ð�Þ; (119)

~ 111ð�Þ ¼ 1

2e4Uð�Þ

�4Q0H

1ð�ÞH2ð�ÞH3ð�Þ

� B2X3a¼1

Pa

Hað�Þ: (120)

From this the components of the charge and moduli de-

pendent Fourier transformed state j ~�ð�Þi are~� lkjð�Þ ¼ e�Uð�Þ ~lkjð�Þ: (121)

Since

j ~�ð�Þi ¼ ðH �H �HÞj�ð�Þi ¼ X1lkj¼0

~�ð�Þjlkji; (122)

one can show that in the asymptotically Minkowski regionwe have


026002-11

lim�!0

j ~�ð�Þi ¼ 1ffiffiffi2

p ðP1j001i þ P2j010i þ P3j100i

� iBðP2 þ P3Þj110i � iBðP1 þ P3Þj101i� iBðP1 þ P2Þj011iþ ½Q0 � B2ðP1 þ P2 þ P3Þj111iÞ: (123)

On the other hand at the horizon we have

lim�!1j ~�ð�Þi ¼ jI4j1=412ðj001i þ j010i þ j100i � j111iÞ:

(124)

This result shows that if we ‘‘start’’ asymptotically with thestate of Eq. (123) with seven nonvanishing genericallydifferent amplitudes we end up with the state ofEq. (124) having merely four nonvanishing amplitudesthat are the same up to a sign. Notice also that the fournonvanishing amplitudes are having states with odd parity.This reminds us of the structure of the fake superpotentialEq. (108). This is as it should be since we know [11] that inthe near horizon limit W 2 should give the square root of�I4 > 0. Indeed, since we have

lim�!1

~�1;2;4ð�Þ ¼ � lim�!1

~�7ð�Þ ¼ 12jI4j1=4; (125)

lim�!1W 2ð�Þ ¼ jI4j1=2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�4Q0P

1P2P3p

which is thecorrect value. According to Eqs. (48) and (52) in ourthree-qubit interpretation

W 2ð1Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�123ðj ~�ð1ÞiÞ

q; (126)

i.e. the square of the fake superpotential on the horizon isjust the entanglement measure of the state of Eq. (124). It isimportant to realize that the components of the fake super-potential are precisely those amplitudes of our three-qubitstate which are not dying out as we are approaching thehorizon. According to our previous results this also worksfor the BPS case (and as will be seen in the next sectioneven for the non-BPS case with vanishing central charge).In order to verify this just recall that for BPS solutions jZjplays the role of the superpotential which is according toEqs. (28) and (64) is again related to the amplitude which isnot dying out in the attractor limit.

VII. ENTANGLED STATES OF GHZ TYPE ON THEHORIZON

In the previous sections we have studied the distillationprocess in the special case of the non-BPS Z � 0 seedsolution. Clearly similar results can be obtained for themost general non-BPS solutions [11] and the non-BPSones with vanishing central charge [8]. In this sectionhowever, our main concern will be to present the explicitforms of our three-qubit state j�ð�Þi at the horizon. Ofcourse the states we expect to show up are again GHZ-likestates, but the new subtlety worth investigating in thiscontext is the appearance of flat directions [37]. As we

have mentioned we have to make a distinction betweenthree different cases of 1

2 -BPS, non-BPS Z ¼ 0, and non-

BPS Z � 0 solutions.In this section we use the pI, qI quantized charges

instead of the PI, QI dressed ones. This is because wewould like to use the most general non-BPS Z � 0 solution[11] which has been produced by using a U-duality trans-formation acting nicely on such quantized charges. Thedressed charges are rescaled quantities related to the quan-tized (undressed) ones via factors coming from the asymp-totic volume moduli. Calculating the moduli ~za using the(15) black hole potential with pI, qI, the asymptotic vol-umes of the tori are nontrivial ~yað0Þ ¼ va. In order to getyað0Þ ¼ 1 as in Eq. (11) we have to rescale the charges withreal positive dressing factors. (For the definitions of thesefactors see the paper by Gimon et al. [31].) The dictionarybetween the two conventions, i.e. pI, qI with ~za ¼ zava

and PI, QI with za is then affected by the correspondence

VBHðPI;QI; zaÞ ¼ GNVBHðpI; qI; ~z

aÞ; (127)

where GN is the D ¼ 4 Newton constant. The quarticinvariant defined in Eq. (49) can also be written in termsof the quantized charges, hence

I4ðPI;QIÞ ¼ G2NI4ðpI; qIÞ: (128)

In this section we denote I4 ¼ I4ðpI; qIÞ. One can checkthat not only the norm of� i.e. the square root of the blackhole potential, but also our three-qubit state scales simplywith

ffiffiffiffiffiffiffiGN

p

j�ðPI;QI; zaÞi ¼ ffiffiffiffiffiffiffi

GN

p j�ðpI; qI; ~zaÞi: (129)

A. The BPS case

After these technicalities first we turn once again to theBPS solutions. The black hole charge configurations sup-porting the 1

2 -BPS attractors at the event horizon are the

ones satisfying the following set of constraints [11]:

I4 > 0; papb � p0qc > 0: (130)

In this case the general 12 -BPS attractor flow solution is

[11,29]

expð�4UÞ ¼ I4ðhI; hIÞ; (131)

~x að�Þ ¼ hIhI � 2haha2ðhbhc � h0haÞ

; (132)

~y að�Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI4ðhI; hIÞ

p2ðhbhc � h0haÞ

: (133)

Here the indices a, b, and c are distinct elements of the setf1; 2; 3g, and no summation is implied on them; on the otherhand, for the indices I ¼ 0, 1, 2, 3 a summation is under-stood. The undressed harmonic functions are defined simi-larly to the (42) dressed ones


026002-12

hIð�Þ ¼ �pI þ pI�; hIð�Þ ¼ �qI þ qI�: (134)

In the horizon limit we have

lim�!1~x

a ¼ pIqI � 2paqa2ðpbpc � p0qaÞ

;

lim�!1~y

a ¼ffiffiffiffiI4

p2ðpbpc � p0qaÞ

:

(135)

In order to obtain j ~�ð�Þi on the horizon, we have to applyðP � P � PÞðS3 � S2 � S1Þ on the charge vector and takethe limit � ! 1. For notational simplicity we introduce theabbreviation

j�i � lim�!1j�ð�Þi: (136)

Then using the identity

4ðp2p3 � p0q1Þðp3p1 � p0q2Þðp1p2 � p0q3Þ¼ ðp0Þ2I4 þ ð2p1p2p3 � p0pIqIÞ2; (137)

we get

j ~�i ¼ �i

2

Ið1=4Þ4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 þ �2

p ½�ðj000i þ j110i þ j101i þ j011iÞ

þ i�ðj111i þ j001i þ j010i þ j100iÞ: (138)

The discrete Fourier transform of this state is

j�i ¼ Ið1=4Þ4

1ffiffiffi2

p�

�� i�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 þ �2

p j000i � �þ i�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 þ �2

p j111i;

(139)

with

� ¼ffiffiffiffiffiffiffijI4j

qp0; � ¼ 2p1p2p3 � p0pIqI; (140)

in accordance with our previous results [3] and Eqs. (64)and (65). It is important to realize at this point that our stateat the horizon can alternatively be written as

j�i ¼ jZj 1ffiffiffi2

p ½e�i argðZÞj000i � ei argðZÞj111i; (141)

where the quantities jZj and argðZÞ now refer to the mag-nitude and phase of the central charge at the horizon.Recall that argðZð�ÞÞ satisfies the following equation:

d

d�argðZð�ÞÞ þAð�Þ ¼ 0; A ¼ 1

2

X3a¼1

dya

ya; (142)

i.e. A is the Kahler connection. Hence the relative phasefactors that show up in the GHZ states of Eqs. (64) and(139) are just the attractor values for the phase of thecentral charge governed by Eq. (142).

B. The non-BPS Z ¼ 0 case

The non-BPS Z ¼ 0 solutions [8,11] can be obtainedfrom the 1

2 -BPS ones simply by changing the sign of any

two imaginary parts of the moduli. This yields the changeof the (130) 1

2 -BPS constraints

I4 > 0; papb � p0qc > 0;

pbpc � p0qa < 0; pcpa � p0qb < 0:(143)

During the calculation of j ~�i the moduli appear only in theSa matrices. We can carry out the sign flip of some ~ya withthe �3 Pauli matrix:

� �3Sa ¼ 1ffiffiffiffiffi~ya

p �~ya 0�~xa 1

� �: (144)

Because of this j ~�23i ¼ ð�3 � �3 � IÞj ~�i and j�23i ¼ðH �H �HÞj ~�23i ¼ ð�1 � �1 � IÞj�i, where the indicesof j�23i denote which moduli have been conjugated. Byvirtue of these observations we get

j ~�12i ¼ �iI1=44

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 þ �2

p ½�ðj000i � j110i � j101i þ j011iÞ

þ i�ðj111i � j001i � j010i þ j100iÞ; (145)

j ~�23i ¼ �iI1=44


p ½�ðj000i þ j110i � j101i � j011iÞ

þ i�ðj111i þ j001i � j010i � j100iÞ; (146)

j ~�13i ¼ �iI1=44


p ½�ðj000i � j110i þ j101i � j011iÞ

þ i�ðj111i � j001i þ j010i � j100iÞ: (147)

The discrete Fourier transform of these states is

j�12i ¼ I1=44

1ffiffiffi2

p�



p j100i;

(148)

j�23i ¼ I1=44

1ffiffiffi2

p�



p j001i;

(149)

j�13i ¼ I1=44

1ffiffiffi2

p�



p j010i:

(150)

Note that � and � are the same as in the 12 -BPS case,

however, now the charge configuration should be compat-ible with the restrictions of Eq. (143).We can also write these states in the form of Eq. (141).

For example, singling out the first qubit we get


026002-13

j�23i ¼ jZ1j 1ffiffiffi2

p ½ei argðZ1Þj110i � e�i argðZ1Þj001i; (151)

where

Za � DaZ ¼ �2iyaDaZ: (152)

For the definition ofDaZ, see Eq. (19). Here as in Eq. (141)the quantities jZ1j and argðZ1Þ refer to their attractorvalues. Generally these quantities are � dependent. Forexample argðZ1ð�ÞÞ satisfies the following equation:

d

d�argðZ1ð�ÞÞ þA1ð�Þ ¼ 0;

A1 ¼ 1

2

�dy1

y1� dy2

y2� dy3

y3

�: (153)

Hence just like the phase of the central charge argðZÞ for12 -BPS solutions for this non-BPS case the phase

� argðZ1ð�ÞÞ flows to a value arctanð�=�Þ as determinedby the expressions in Eq. (140). The important differencehere is the fact that in this case we have a different chargeconfiguration which should now respect the non-BPS con-straints of Eq. (143). Clearly, similar results hold for thesecond and third qubits playing a special role. Notice alsothat the quantities jZaj where a ¼ 1, 2, 3 occurring in theexpressions of the three-qubit states like Eqs. (141) and(151) are just the attractor values of the fake superpotential

W að�Þ ¼ jZað�Þj: (154)

For the 12 -BPS case a similar role is played by the quantity

W ð�Þ � jZð�Þj. SinceZð�Þ ¼ � ffiffiffi

2p

�111; Z1 ¼ffiffiffi2

p�110;

Z2ð�Þ ¼ffiffiffi2

p�101; Z3 ¼

ffiffiffi2

p�011

(155)

with the remaining amplitudes arising by complex conju-gation [see Eq. (30)] we see that the fake superpotential inthe relevant cases is related to the magnitudes of thoseamplitudes which are not dying out as the correspondingBPS or non-BPS flow approaches the horizon.

C. The non-BPS Z � 0 case

The general non-BPS Z � 0 case [11] is extremelydifferent. The general attractor flow solution is

expð�4Uð�ÞÞ ¼ h0ð�Þh1ð�Þh2ð�Þh3ð�Þ � b2;

~xað�Þ ¼ &a�2aC

a1 þ ð&a � %aÞ�aC

a2 � %aC

a3

�2aC

a1 þ 2�aC

a2 þ Ca

3

;

~yað�Þ ¼ ð&a þ %aÞ2�aC4

�2aC

a1 þ 2�aC

a2 þ Ca

3

; (156)

where

�a ¼ �e�a ; � ¼��þ �

��

�1=3

; (157)

&a ¼ffiffiffiffiffiffiffiffiffi�I4

p þ ðpIqI � 2paqaÞ2ðpbpc � p0qaÞ

; (158)

%a ¼ffiffiffiffiffiffiffiffiffi�I4

p � ðpIqI � 2paqaÞ2ðpbpc � p0qaÞ

; (159)

are charge-dependent constants with � and � given byEq. (140). The �a real constants satisfying the constraint

�1 þ �2 þ �3 ¼ 0; (160)

account for the flat directions [11,37]. The harmonic func-tions now defined as

hIð�Þ ¼ bI þ ð�I4Þð1=4Þ�; (161)

giving rise to the quantities

Ca1 ¼ hbhc þ h0ha þ 2b; (162)

Ca2 ¼ hbhc � h0ha; (163)

Ca3 ¼ hbhc þ h0ha � 2b; (164)

C4 ¼ expð�2UÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih0h1h2h3 � b2

q; (165)

also making their presence in Eqs. (156).One can obtain the non-BPS seed solution [31] inves-

tigated in Secs. V and VI as a special case of the general

non-BPS Z � 0 solution with the parameters &a ¼ %a ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�q0pa=pbpc

p, � ¼ 1 ¼ �a, �a ¼ 0. Here b ¼ B=GN is

the undressed version of the B field of the seed solution.Now we can present the horizon limit of�, where some

calculational steps needed for its derivation are left for theAppendix

j ~�i ¼ 1

2

ð�I4Þ1=4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�2 � �2Þsgnð�Þ coshð�3 þ ’Þ coshð�2 þ ’Þ coshð�1 þ ’Þp ½ 0j000i þ 1j110i þ 2j101i þ 3j011i

þ �0j111i þ �1j001i þ �2j010i þ �3j100i; (166)


026002-14

where

0 ¼ �i�;

a ¼ ið�� sinhð��a þ 2’Þ þ � coshð��a þ 2’ÞÞ;(167)

�0 ¼ ��;

�a ¼ sgnð�Þð� coshð�a þ ’Þ � � sinhð�a þ ’ÞÞ; (168)

and let � ¼ sgnð�Þe’:’ ¼ lnj�j: (169)

Let us now consider some special charge configurationsgiving rise to non-BPS Z � 0 solutions. For the D0�D4configuration only the charges q0 and pa are switched on.We consider the case when q0 < 0 and pa > 0. Then I4 ¼4q0p

1p2p3 < 0, � ¼ 0, � ¼ 2p1p2p3 > 0, � ¼ 1, and

j ~�i ¼ ð�4q0p1p2p3Þ1=4

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficoshð�3Þ coshð�2Þ coshð�1Þ

p ½i sinhð�1Þj110i

þ i sinhð�2Þj101i þ i sinhð�3Þj011i � j111iþ coshð�1Þj001i þ coshð�2Þj010iþ coshð�3Þj100i: (170)

As a special case when �1 ¼ �2 ¼ �3 ¼ 0 one gets

j ~�i ¼ ð�4q0p1p2p3Þ1=412½j001i þ j010i þ j100i � j111i;

(171)

i.e. we get back to the state at the horizon obtained for theseed solution in Eq. (124).

Now we consider the dual case of the D2�D6 chargeconfiguration [11,32,38]. After choosing p0 > 0, qa > 0,

I4 ¼ �4p0q1q2q3 < 0, � ¼ ffiffiffiffiffiffiffijI4jp

p0 > 0, � ¼ 0, and � ¼�1, one obtains

j ~�i ¼ ð4p0q1q2q3Þð1=4Þ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficoshð�3Þ coshð�2Þ coshð�1Þ

p ½�ij000i

þ i coshð�1Þj110i þ i coshð�2Þj101iþ i coshð�3Þj011i þ sinhð�1Þj001iþ sinhð�2Þj010i þ sinhð�3Þj100i: (172)

Specially, when �1 ¼ �2 ¼ �3 ¼ 0

j ~�i ¼ ið4p0q1q2q3Þ1=412½j110i þ j101i þ j011i � j000i:(173)

Comparing Eqs. (171) and (173) one can see that forvanishing flat directions the D0�D4 amplitudes are realand the D2�D6 ones are purely imaginary. Moreover,these cases are dual in the sense that they are related by thebit-flip operation �1 � �1 � �1. Notice also that the norms

of these states giveffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�4q0p

1p2p3p

andffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4p0q1q2q3

papart

from the fact whether the flat directions are vanishing or

not. These quantities multiplied by � give the macroscopicblack hole entropy [7].It is interesting to analyze the effect of the asymptotic

data on such states on the horizon. More precisely we areinterested in those changes that leave the entropy (i.e. thenorm of the state) invariant. As an example let us considerthe D2�D6 case. In the hope to have an effect merely onthe relative phases of the state at the horizon we can adjustthe asymptotic values for the charges and the �a parame-ters (flat directions) coming from the moduli. Other infor-mation coming from the asymptotic moduli is swallowedby the attractor mechanism.Let us change the signs of the charges q1, q2, and q3 in

such a way that the constraint p0q1q2q3 > 0 is notchanged. Then one can show [7] that the possibilities for

j ~�i arej ~�im3m2m1

¼ ið4p0q1q2q3Þ1=412½m1j110i þm2j101iþm3j011i � j000i; (174)

where

ðm3; m2; m1Þ 2 fðþ þþÞ; ðþ ��Þ; ð� þ�Þ; ð� �þÞg:(175)

Hence although these changes are not affecting the blackhole entropy, they have an effect on the particular form ofthe state. As one can check the possible changes in signgiving rise to the four states of Eq. (174) can be representedby phase flip error operators as

�3 � �3 � I; �3 � I � �3; I � �3 � �3; (176)

where for example

j ~�i��þ ¼ ð�3 � �3 � IÞj ~�iþþþ: (177)

Alternatively we can apply the corresponding bit-flip erroroperators containing �1 on the Fourier transformed states.Notice that these four states are all invariant under �3 ��3 � �3. The result of this is that the subspace spanned bythese states is invariant under an arbitrary number of phaseflips (or bit flips for the Fourier transformed subspace.) Wecan thus conclude that the effect of the change of thisparticular type of asymptotic data on the state at thehorizon is the appearance of phase or bit-flip errors asso-ciated with an invariant subspace spanned by the states ofEq. (174).Let us now remain in the non-BPS Z � 0 charge orbit

and fix the signs of the D2�D6 charges, but this time letus change the asymptotic values for the parameters of theflat directions from �a � 0 to �a � 0, �1 þ �2 þ �3 ¼ 0.In this case we see that the uniform structure of Eq. (173)deteriorates via the schematic transformation rule (neglect-ing the normalization factor)

j000i � j000i;j110i � cosh�1j110i � i sinh�1j001i; etc:

(178)


026002-15

Let us denote this new state i.e. the one of Eq. (172) by

j ~�i�3�2�1. Then one can show that

j ~�i�3�2�1¼ ðE3 � E2 � E1Þj ~�iþþþ; (179)

where

Ea � 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffifficosh�a

p 1 0

i sinh�a cosh�a

!;

�1 þ �2 þ �3 ¼ 0: (180)

Hence the changes on the state j ~�iþþþ originating fromthe flat directions have the obvious interpretation of errorsof a more general kind depending on continuously chang-ing parameters. Notice that we would have obtained thesame state after changing the sign of the term cosh�a in thelower right corner of the matrix in Eq. (180). Such matricesE�a 2 GLð2;CÞ in the limit �a ! 0 result in the phase flip

error operators �3 acting on the corresponding qubit wehave already discussed. In quantum information theory theGLð2;CÞ operators acting on the qubits are called trans-formations associated with SLOCC [14]. It is amusing tosee that though the error operators E�

a act locally but theconstraint �1 þ �2 þ �3 ¼ 0 (in the type IIA dualityframe coming from deformations preserving the overallvolume of T6) refers to the fact that they are not indepen-dent. In quantum information theory such constraints usu-ally refer to an agreement between the parties affected viathe use of classical channels. Finally in closing this sectionwe note that the normalized part of the attractor state of

Eq. (173) i.e. j ~�iþþþ is just the one which can be used toestablish a very striking version of Bell’s theorem [48].

D. The D0�D6 case

As another special subcase of the non-BPS Z � 0 one,finally we consider the D0�D6 solution. This chargeconfiguration can only appear in the non-BPS regime,because I4 ¼ �ðp0q0Þ2 < 0 independent of the signs ofthe charges. Originally, the general non-BPS Z � 0 solu-tion was produced by an SLð2;RÞ�3 U-duality transforma-tion of the D0�D6 one [11,31]. Because of the &a, %a

parametrization of this transformation [see Eqs. (158) and(159)], we cannot produce either the identity transforma-tion or the transformations that bring us back to the D0�D6 case with different charges. Hence we cannot simplywrite the D0�D6 charges into the corresponding formu-las for the horizon limit of �ð�Þ. However, for the calcu-lation of �ð�Þ on the horizon we can directly use theoriginal D0�D6 solutions [11] instead.

expð�4Uð�ÞÞ ¼ h0ð�Þh1ð�Þh2ð�Þh3ð�Þ � b2; (181)

~x að�Þ ¼ �0aC

a2

Ca3

; (182)

~y að�Þ ¼ 2�0aC4

Ca3

; (183)

with the notation introduced in Eqs. (162)–(165), and

�0a ¼

�q0p0

�1=3

e�a : (184)

Note that on the horizon xa ! 0 and ya ! �0a. With this

moduli the calculation of � is much easier than in thegeneral case. Finally a straightforward calculation yieldsthe result

j ~�i ¼ 1ffiffiffi2

p ð�I4Þ1=4½�ij000i þ j111i: (185)

Note that this state is independent of the �a parameters offlat directions.

VIII. CONCLUSIONS

In this paper we have shown that in the special case ofthe STU model the attractor mechanism for extremal,static, and spherically symmetric BPS and non-BPS blackhole solutions can be cast in a form of a distillationprocedure of entangled three-qubit states of a specialkind on the horizon. Such states are belonging to the so-called GHZ class featuring maximum tripartite entangle-ment [14]. In obtaining this result our main calculationaltool was to organize the charges, the moduli fields, and thewarp factor in a suitably defined three-qubit state[Eqs. (25), (26), and (76)]. This state is just lying on theGLð2;CÞ�3 orbit of a three-qubit ‘‘charge state’’ [Eq. (22)]fixing the charge orbit to which the particular solutionbelongs. To conform with the well-known fact that theduality group of the STU model is not GLð2;CÞ�3 butSLð2;RÞ�3, our state is also satisfying a reality condition[Eq. (31)]. On this three-qubit state the flat covariantderivatives are acting as phase (sign) or bit-flip errors[Eq. (34)] depending on whether we express the state inthe computational basis or in its discrete Fourier trans-formed version [Eqs. (38) and (39)]. The black hole po-tential can be expressed as the norm of our state [Eq. (27)].For spherically symmetric solutions such states are also

displaying an explicit dependence on the radial coordinater ¼ 1

� referring to the distance from the horizon. By solving

the equations of motion for the moduli fields and warpfactor we end up with a flow of BPS or non-BPS typedepending on the charge configuration. Using the explicitforms of these flows that have recently become available inthe literature we can study the distillation procedure indetail. In order to illustrate how this distillation becomesunfolded as we approach the horizon we have chosen therecently discovered non-BPS seed solution.For such solutions we observed that the charge, moduli,

and warp factor dependent state of Eq. (76) satisfies asystem of first order differential equations [Eqs. (102) and


026002-16

(109)] featuring the fake superpotential [Eq. (105)]. Thisobservation conforms with the recent results on the firstorder formalism relating supergravity flows to geodesicmotion on the moduli space of the 3D dimensionallyreduced theory [33,47]. In light of this connection in anaccompanying paper we elaborate further on this point andestablish an entanglement based understanding of some ofthese results [35].

For the non-BPS seed solutions we managed to demon-strate how a standard GHZ state at the horizon emergesfrom a state characterizing the flow at the asymptoticallyMinkowski region. The attractor mechanism in this picturesimply amounts to the fact that three amplitudes out of theseven nonequal nonvanishing ones of our three-qubit stateare dying out as we approach the horizon [see Eqs. (123)and (124)]. The remaining amplitudes have the same mag-nitudes related to the macroscopic black hole entropy. Therelative phase factors of these amplitudes are merely signsreflecting the structure of the fake superpotential [11,33].

In this paper we also conducted a detailed study on thestructure of ‘‘attractor states.’’ By this term we denote theparticular states we obtain from our � dependent ones afterperforming the � ! 1 limit. We have shown that the12 -BPS and non-BPS Z ¼ 0 cases are very similar. The

attractor states are of canonical GHZ form with the relativephases related to the phases of the central charge Z or thephases of the quantities Za, a ¼ 1, 2, 3 which are just theflat covariant derivatives of Z [Eqs. (141) and (151)]. Weobserved that in these cases and also in the case of the Z �0 seed solutions the fake superpotential is related to thoseamplitudes of our � dependent states which are not dyingout as we approach the horizon.

The new subtlety arising in the non-BPS Z � 0 case isthe appearance of flat directions. As it is known for theD0�D6 case the parameters labeling the flat directionsare related to deformations of the volumes of the three toriT2 preserving the overall volume of T6 (in the type IIAduality frame). We have found that for this charge configu-ration the flat directions are not making their presence inthe corresponding attractor state. (Though they do appearin the particular form of the attractor values of the moduli.)However, for the most general charge configuration flatdirections do appear in the attractor states. In the specialcases of the D0�D4 and D2�D6 systems we haveshown that the effect of the flat directions is to deterioratethe uniform structure of the corresponding attractor statesobtained by starting the flow not in any of the flat direc-tions. It is known [7] that the effect of changing the signs ofthe D2 and D4 charges asymptotically results in phase orbit-flip errors on the attractor states. By virtue of this thepresence of flat directions adds an additional twist to thispicture. In particular we have demonstrated that flat direc-tions can entertain the possible interpretation as errors of amore general type (i.e. ones depending on continuouslychanging parameters) acting on attractor states.

Now we comment on the possible physical relevance ofour three-qubit states. Obviously our compressing of thevarious ingredients of the STU model in a three-qubit stateat this stage is merely a nice way of understanding thestructure of BPS and non-BPS solutions in the STU model.Notice, however, that the attractor states are always just theones that are connected to the structure of the fake super-potential. Indeed, the fake superpotential in our examplesturned out to be related to those amplitudes of our three-qubit states which are not dying out during the process ofdistillation. It is known that upon quantization of the radialevolution of the moduli [34] results in a semiclassical wavefunction the phase of which is featuring the quantity eUWwhich is just formed out of the aforementioned amplitudesof our three-qubit state of Eq. (76). [See also Eqs. (105),(107), and (108).] This observation might give a cluetoward getting a deeper insight into the physical meaningof our GHZ-like states.We also would like to stress that apart from establishing

a dictionary between results from one side of the blackhole-qubit correspondence in the language of the other, ourcharge, moduli, and warp factor dependent three-qubitstates also proved to be useful for hinting at a basic four-qubit picture of STU black holes hiding behind the scenes.Indeed, one of the aims of the present paper (via expressingas much data as we can in terms of entities of QIT) was alsoto set the stage for our next paper elucidating this under-lying four-qubit reformulation [35]. The key observation inthis respect is the well-known fact that stationary solutionsin D ¼ 4 supergravity can be elegantly described by di-mensional reduction along the time direction [49]. In thispicture stationary solutions can be identified as solutions toa D ¼ 3 nonlinear sigma model with target space being asymmetric space G=H with H noncompact. The propertythat is of basic significance is that the group G in this caseextends the global symmetry group G4 of D ¼ 4 super-gravity, by also incorporating the Ehlers [50] SLð2;RÞ. Inour specific case the N ¼ 2 STU model can be regarded asa consistent truncation of maximal N ¼ 8, D ¼ 4 super-gravity with G4 ¼ E7ð7Þ, truncating to SLð2;RÞ�3.

Timelike reduction in the general case then yields[47,51–54] the coset E8ð8Þ=SO�ð16Þ, or in the case of the

STU truncation the one M3 ¼ SOð4; 4Þ=SLð2;RÞ�4.Hence it is natural to suspect that for stationary solutionsobjects like jð�Þi of Eq. (76) are really four-qubit states ina disguised form with the Ehlers group acting on a hiddenextra qubit. Expressing the basic data in terms of quantitiesof QITas we initiated in this paper and also incorporating afourth qubit via the Ehlers group STU black holes canindeed be described as entangled four-qubit states [35].We also suggested [35] that the four-qubit entanglementclasses can be mapped to the nilpotent orbits [51] used forthe classification of extremal static spherically symmetricSTU black hole solutions. This challenge recently wastaken up by Borsten et al. [55] verifying our claim.


026002-17

Moreover, since the current literature on the classificationproblem of four-qubits in QIT contains seemingly contra-dictory results, the input from string theory might settlethis issue.

Finally notice that although being very special, the STUmodel captures the essential features also of extremal blackholes in the N ¼ 4, 8 theories. Moreover many of itsfeatures generalize well to other black hole solutions(such as those arising from Calabi-Yau compactifications).In this respect we just remark that the maximalN ¼ 8, d ¼4 supergravity has seven STU subsectors corresponding toits consistent truncations. In the corresponding extremalblack hole solution context this observation has alreadybeen related to systems exhibiting tripartite entanglementof seven qubits [4,5]. It would be interesting to studydistillation issues for this more general scenario using theideas as developed in this paper.

APPENDIX: CALCULATING �ð�Þ ON THEHORIZON (NON-BPS Z � 0 CASE)

In this Appendix we outline the main steps leading to theexplicit expression of � on the horizon. First recall theexplicit form of the non-BPS Z � 0 attractor flow ofEqs. (156)–(165). Using the (158) and (159) forms of &aand %a, we can alternatively write this flow as

~x að�Þ ¼ pIqI � 2paqa2ðpbpc � p0qaÞ

þffiffiffiffiffiffiffiffiffi�I4

p2ðpbpc � p0qaÞ

Caxð�Þ;

(A1)

~y að�Þ ¼ffiffiffiffiffiffiffiffiffi�I4

p2ðpbpc � p0qaÞ

Cayð�Þ: (A2)

Here the � dependent terms are

Caxð�Þ ¼ �2

aCa1 � Ca

3

�2aC

a1 þ 2�aC

a2 þ Ca

3

; (A3)

Cayð�Þ ¼ 4�aC4

�2aC

a1 þ 2�aC

a2 þ Ca

3

: (A4)

Since the moduli can be written as ~xað�Þ ¼ Aa þ BaCaxð�Þ

and ~yað�Þ ¼ BaCayð�Þ, the transformation Sa of Eq. (25)

can be expressed as

Sa ¼ 1ffiffiffiffiffiya

p CaBaAa; (A5)

where

C a ¼ Cay 0

�Cax 1

� �; Ba ¼ Ba 0

0 1

� �; Aa¼ 1 0

�Aa 1

� �:

(A6)

After these preliminaries the transformation S3 � S2 �S1 can be carried out in three steps. First in order to use the(140) definition of � and � also in this non-BPS case, theidentity (137) has to be changed as

4ðp2p3 � p0q1Þðp3p1 � p0q2Þðp1p2 � p0q3Þ ¼ �2 � �2

(A7)

due to I4ð�Þ< 0. By virtue of this we can perform the firsttwo transformations

ðB3A3 �B2A2 �B1A1Þj�i

¼ 1ffiffiffi8

p �I4�2 � �2

½�ðj000i þ j011i þ j101i þ j110iÞ

þ �ðj111i þ j100i þ j010i þ j001iÞ

¼ �I4�2 � �2

�1

2ð�þ �Þj~0 ~0 ~0i � 1

2ð�� Þj~1 ~1 ~1i

;

(A8)

where j~0i and j~1i are the (39) Hadamard transformedstates. On this latter form the transformation ðC3 � C2 �C1Þ acts readily. The � dependency appears only in Ca. Inthe horizon limit we obtain

lim�!1C

ax ¼ �2

a � 1

�2a þ 1

¼12 ð�a � 1

�aÞ

12 ð�a þ 1

�aÞ ;

lim�!1C

ay ¼ 2�a

�2a þ 1

¼ 112 ð�a þ 1

�aÞ ;

(A9)

and with these formulas we get the result of (166) and(167).

[1] M. J. Duff, Phys. Rev. D 76, 025017 (2007).[2] R. Kallosh and A. Linde, Phys. Rev. D 73, 104033 (2006).[3] P. Levay, Phys. Rev. D 74, 024030 (2006).[4] M. J. Duff and S. Ferrara, Phys. Rev. D 76, 025018 (2007).[5] P. Levay, Phys. Rev. D 75, 024024 (2007).[6] M. J. Duff and S. Ferrara, Phys. Rev. D 76, 124023 (2007).[7] P. Levay, Phys. Rev. D 76, 106011 (2007).

[8] S. Bellucci, A. Marrani, E. Orazi, and A. Shcherbakov,Phys. Lett. B 655, 185 (2007).

[9] L. Borsten, D. Dahanayake, M. J. Duff, W. Rubens, and H.Ebrahim, Phys. Rev. Lett. 100, 251602 (2008).

[10] L. Borsten, Fortschr. Phys. 56, 842 (2008); L. Borsten, D.Dahanayake, M. J. Duff, H. Ebrahim, and W. Rubens,Phys. Rev. A 80, 032326 (2009); L. Borsten, D.


026002-18








http://dx.doi.org/10.1016/j.physletb.2007.08.079

http://dx.doi.org/10.1103/PhysRevLett.100.251602

http://dx.doi.org/10.1002/prop.200810542

http://dx.doi.org/10.1103/PhysRevA.80.032326

Dahanayake, M. J. Duff, and W. Rubens, Phys. Rev. D 80,026003 (2009).

[11] S. Bellucci, S. Ferrara, A. Marrani, and A. Yerayan,Entropy 10, 507 (2008).

[12] L. Borsten, D. Dahanayake, M. J. Duff, H. Ebrahim, andW. Rubens, Phys. Rep. 471, 113 (2009).

[13] P. Levay, M. Saniga, and P. Vrana, Phys. Rev. D 78,124022 (2008).

[14] W. Dur, G. Vidal, and J. I. Cirac, Phys. Rev. A 62, 062314(2000).

[15] P. Levay and P. Vrana, Phys. Rev. A 78, 022329 (2008).[16] P. Vrana and P. Levay, J. Phys. A 42, 285303 (2009).[17] D. Gottesman, Phys. Rev. A 54, 1862 (1996); 57, 127

(1998); A. R. Calderbank, E. E. Rains, P.W. Shor, andN. J. A. Sloane, Phys. Rev. Lett. 78, 405 (1997).

[18] P. Levay, M. Saniga, P. Vrana, and P. Pracna, Phys. Rev. D79, 084036 (2009).

[19] M. Saniga and M. Planat, Advanced Studies in TheoreticalPhysics 1, 1 (2007).

[20] M. Saniga and M. Planat, Quantum Inf. Comput. 8, 0127(2008).

[21] N. D. Mermin, Phys. Rev. Lett. 65, 3373 (1990); Rev.Mod. Phys. 65, 803 (1993); A. Peres, J. Phys. A 24,L175 (1991).

[22] M. Planat and P. Sole, J. Phys. A 42, 042003 (2009).[23] B. L. Cherciai and B. Van Geemen, arXiv:1003.4255.[24] M. J. Duff, J. T. Liu, and J. Rahmfeld, Nucl. Phys. B459,

125 (1996).[25] K. Behrndt, R. Kallosh, J. Rahmfeld, M. Shmakova, and

W.K. Wong, Phys. Rev. D 54, 6293 (1996).[26] S. Ferrara, R. Kallosh, and A. Strominger, Phys. Rev. D

52, R5412 (1995); S. Ferrara and R. Kallosh, Phys. Rev. D54, 1514 (1996); R. Kallosh, Phys. Rev. D 54, 1525(1996); A. Strominger, Phys. Lett. B 383, 39 (1996); S.Ferrara, G.W. Gibbons, and R. Kallosh, Nucl. Phys. B500,75 (1997).

[27] D.M. Greenberger, M. Horne, and A. Zeilinger, Bell’sTheorem, edited by M. Kafatos (Kluwer, Dordrecht,1989), p. 69.

[28] M. Hein, W. Dur, J. Eisert, R. Raussendorf, M. Van denNest, and H. J. Briegel, arXiv:quant-ph/0602096.

[29] K. Behrndt, D. Lust, and W.A. Sabra, Nucl. Phys. B510,264 (1998); W.A. Sabra, Nucl. Phys. B510, 247 (1998);Mod. Phys. Lett. A 12, 2585 (1997).

[30] K. Hotta and T. Kubota, Prog. Theor. Phys. 118, 969(2007).

[31] E. G. Gimon, F. Larsen, and J. Simon, J. High EnergyPhys. 01 (2008) 040.

[32] Rong-Gen Cai and Da-Wei Pang, J. High Energy Phys. 01(2008) 046.

[33] D. Z. Freedman, C. Nunez, M. Schnabl, and K. Skenderis,Phys. Rev. D 69, 104027 (2004); A. Celi, A. Ceresole, G.Dall’Agata, A. Van Proeyen, and M. Zagerman, Phys. Rev.

D 71, 045009 (2005); A. Ceresole and G. Dall’Agata, J.High Energy Phys. 03 (2007) 110.

[34] M. Gunaydin, A. Neitzke, B. Pioline, and A. Waldron, J.High Energy Phys. 09 (2007) 056; Phys. Rev. D 73,084019 (2006); G. Bossard, Y. Michel, and B. Pioline, J.High Energy Phys. 01 (2010) 038.

[35] P. Levay, arXiv:1004.3639.[36] S. Ferrara, E. G. Gimon, and R. Kallosh, Phys. Rev. D 74,

125018 (2006).[37] S. Nampuri, P. K. Tripathy, and S. Trivedi, J. High Energy

Phys. 08 (2007) 054; S. Ferrara and A. Marrani, Phys.Lett. B 652, 111 (2007).

[38] R. Kallosh, N. Sivanandam, and M. Soroush, Phys. Rev. D74, 065008 (2006).

[39] A. Strominger, Commun. Math. Phys. 133, 163 (1990); A.Ceresole, R. D’Auria, and S. Ferrara, Nucl. Phys. B, Proc.Suppl. 46, 67 (1996).

[40] R. Kallosh, S. Srivanandam, and M. Soroush, J. HighEnergy Phys. 03 (2006) 060.

[41] F. Denef, J. High Energy Phys. 08 (2000) 050; B. Batesand F. Denef, arXiv:hep-th/0304094.

[42] M. Cvetic and D. Youm, Phys. Rev. D 53, R584 (1996).[43] R. Emparan and G. T. Horowitz, Phys. Rev. Lett. 97,

141601 (2006).[44] A. Cayley, Camb. Math. J. 4, 193 (1845); I.M. Gel’fand,

M.M. Kapranov, and A.V. Zelevinsky, Discriminants,Resultants and Multidimensional Determinants(Birkhauser, Boston, 1994).

[45] V. Coffman, J. Kundu, and W.K. Wootters, Phys. Rev. A61, 052306 (2000).

[46] J.-G. Luque and J.-Y. Thibon, Phys. Rev. A 67, 042303(2003); P. Levay, J. Phys. A 39, 9533 (2006).

[47] G. Bossard, H. Nicolai, and K. S. Stelle, J. High EnergyPhys. 07 (2009) 003.

[48] D. Mermin, Quantum Computer Science: An Introduction(Cambridge University Press, Cambridge, England, 2007),Chap. 6.6.

[49] P. Breitenlohner, D. Maison, and G.W. Gibbons,Commun. Math. Phys. 120, 295 (1988); P. Breitenlohnerand D. Maison, Commun. Math. Phys. 209, 785 (2000).

[50] J. Ehlers, Ph.D. thesis, Hamburg University, 1957; R.Geroch, J. Math. Phys. (N.Y.) 12, 918 (1971).

[51] G. Bossard, Y. Michel, and B. Pioline, J. High EnergyPhys. 01 (2010) 038.

[52] G. Bossard, H. Nicolai, and K. S. Stelle, Gen. Relativ.Gravit. 41, 1367 (2009).

[53] M. Gunaydin, A. Neitzke, B. Pioline, and A. Waldron, J.High Energy Phys. 09 (2007) 056.

[54] E. Bergshoeff, W. Chemissany, A. Ploegh, M. Trigiante,and T. Van Riet, Nucl. Phys. B812, 343 (2009).

[55] L. Borsten, D. Dahanayake, M. J. Duff, A. Marrani, andW. Rubens, arXiv:1005.4915.


026002-19



http://dx.doi.org/10.3390/e10040507

http://dx.doi.org/10.1016/j.physrep.2008.11.002






http://dx.doi.org/10.1088/1751-8113/42/28/285303








http://dx.doi.org/10.1103/RevModPhys.65.803

http://dx.doi.org/10.1103/RevModPhys.65.803

http://dx.doi.org/10.1088/0305-4470/24/4/003

http://dx.doi.org/10.1088/0305-4470/24/4/003

http://dx.doi.org/10.1088/1751-8113/42/4/042003

http://arXiv.org/abs/1003.4255

http://dx.doi.org/10.1016/0550-3213(95)00555-2

http://dx.doi.org/10.1016/0550-3213(95)00555-2


http://dx.doi.org/10.1103/PhysRevD.52.R5412






http://dx.doi.org/10.1016/0370-2693(96)00711-3

http://dx.doi.org/10.1016/S0550-3213(97)00324-6

http://dx.doi.org/10.1016/S0550-3213(97)00324-6

http://arXiv.org/abs/quant-ph/0602096

http://dx.doi.org/10.1016/S0550-3213(97)00633-0

http://dx.doi.org/10.1016/S0550-3213(97)00633-0

http://dx.doi.org/10.1016/S0550-3213(97)00607-X

http://dx.doi.org/10.1142/S0217732397002715

http://dx.doi.org/10.1143/PTP.118.969

http://dx.doi.org/10.1143/PTP.118.969

http://dx.doi.org/10.1088/1126-6708/2008/01/040

http://dx.doi.org/10.1088/1126-6708/2008/01/040

http://dx.doi.org/10.1088/1126-6708/2008/01/046

http://dx.doi.org/10.1088/1126-6708/2008/01/046




http://dx.doi.org/10.1088/1126-6708/2007/03/110

http://dx.doi.org/10.1088/1126-6708/2007/03/110

http://dx.doi.org/10.1088/1126-6708/2007/09/056

http://dx.doi.org/10.1088/1126-6708/2007/09/056



http://dx.doi.org/10.1007/JHEP01(2010)038





http://dx.doi.org/10.1088/1126-6708/2007/08/054

http://dx.doi.org/10.1088/1126-6708/2007/08/054





http://dx.doi.org/10.1007/BF02096559

http://dx.doi.org/10.1016/0920-5632(96)00008-4

http://dx.doi.org/10.1016/0920-5632(96)00008-4

http://dx.doi.org/10.1088/1126-6708/2006/03/060

http://dx.doi.org/10.1088/1126-6708/2006/03/060

http://dx.doi.org/10.1088/1126-6708/2000/08/050

http://arXiv.org/abs/hep-th/0304094








http://dx.doi.org/10.1088/0305-4470/39/30/009

http://dx.doi.org/10.1088/1126-6708/2009/07/003

http://dx.doi.org/10.1088/1126-6708/2009/07/003

http://dx.doi.org/10.1007/BF01217967

http://dx.doi.org/10.1007/s002200050038

http://dx.doi.org/10.1063/1.1665681



http://dx.doi.org/10.1007/s10714-008-0720-7

http://dx.doi.org/10.1007/s10714-008-0720-7

http://dx.doi.org/10.1088/1126-6708/2007/09/056

http://dx.doi.org/10.1088/1126-6708/2007/09/056

http://dx.doi.org/10.1016/j.nuclphysb.2008.10.023


attractor mechanism as a distillation procedure

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